262 APPENDIX TO CHAPTER IX
(In both cases, define an increasing sequence of bounded components /3(Hn),
where Hn is a simple closed curve, such that the union of the /2(HJ is the given open
set. To prove that fr(AO is not homeomorphic to U, show that it is not locally connected,
using (3.19.1); to prove the similar property for fr(A2) consider the complement of the
point z = I in that set.) Are fr(Ai) and fr(A2) homeomorphic ?
6. Let A be a simply connected open subset of C, distinct from C. Show that the frontier
of A contains at least two distinct points. (Show that otherwise, one would have
A = C - {a}, using (3.19.9) and the fact that C -{a} is connected to prove that there
can be no exterior point of A; conclude by using (9.8.4) and (9.8.7).)
7. Let y be a simple loop defined in I = [0, 2], and let H = y(I) be the corresponding simple
closed curve. Let a be a simple path defined in I2 = [1, 2] and such that: (1) a(l) = y(l),
a(2) = y(2) = y(0); (2) a(t) e j8(H) for every t e ]1, 2[; let L = a(I2). Define the simple
loops yi»y2 in I by the conditions:
yi(r) = y« for 0 < / < 1 , yiM = a(f ) for 1 ^ t ^ 2 ;
y2(/) = a(2 - /) for 0 < t < 1 , y2« = y (0 for 1 ^ t ^ 2.
(a) Show that for any z e C which does not belong to Hj u H2 , j(z; y) «
/fcyi) +yfr;y2) (use (Ap.1.1)).
(b) Prove that there are points zx e j8(H) such that yfo ; yi) = 0, and points z2 e j8(H)
such thaty(z2;y2) = 0 (use (b) of the Jordan curve theorem (Ap.4.2)).
(c) Deduce from (a) and (b) that jB(H) is the union of /?(Hi), /?(H2) and L o j8(H),
no two of these sets having common points.
8. (a) v Let H be a simple closed curve, Lfc (1 ^ k ^ n) n simple arcs, having their ex-
tremities in H, and whose points distinct from the extremities are in /2(H). Suppose in
addition that no two of the L* have common points belonging to j8(H). Then the
n _
interior of the complement of (J Lfc in j8(H) has n -f 1 connected components, each
of which is the bounded component of the complement of a simple closed curve in C.
(Use induction on n, and Problem 7.)
(b) Let HI, H2 be two simple closed curves in C, such that H* n H2 is finite. Show
that each connected component of j&(Hi) n /?(H2) is the bounded component of the
complement in C of a simple closed curve (use (a)).
9. Let y be a simple loop in C, defined in I = [— 1, 1], let H = y(I), and suppose (for
simplicity's sake) thaty(0)=0 and the diameter of H is >2. Define inductively two
decreasing sequences of numbers, (afl) and (/>„), tending to 0, such that pi = 1, afl is
the largest number >0 such that |y(0| < pn for |r| < an, and pn+i = inf | - -, 8H I,
\n -f 1 I
where 8n is the distance of 0 to the set of points y(t) such that \t\ ^ art .
(a) Prove that if z, zf are two points of /?(H) such that \z\ < />rt+1 and \z'\ <pn+i,
then there is a path of extremities z, z', contained in the intersection of /?(H) and of
the closed disk of center 0 and radius pn . (Let L be a simple broken line of extremities
z, z', contained in jS(H) (Problem 1). Suppose first that the segment S of extremities
z, z' has no point in common with L, distinct from z, z'; then R = L u S is a simple
closed curve. Prove that if t el is such that y(f) e/?(R), then \t\<ocn: observe that
the intersection R n H is contained in S, and show that if there was a t e I such that
y(/) e £(R) and |/| ^ an, there would be another t' e I such that y(t') e S and \t'\ ^ a«,
contradicting the definition of pn+l. Conclude in that case by taking the connected
component of the intersection of /?(R) and the open disk of center 0 and radius />„,